And I would seriously recommend taking that double dogleg through Santa Fe and Albuquerque. After all it too is, or was a part of 66, is very scenic and certainly more pleasant than a freeway.
I took this route, and with a couple of exceptions had no problems. And Route 66 has many more accommodations BECAUSE of the freeway. Nome backwater road that no oe other than local residences use will be guaranteed desolate.